Arrangements and Groups, 119 



The arrangements are the following : 

 aa ab ac ad 

 ba bb be bd 

 ca cb cc cd 

 da db dc dd 



Thus we see that there are sixteen arrangements, that is, ^ 

 arrangements. In exactly the same way if we have n letters 

 a, b, c, d, e,f, . . . , the a may be followed by each of the 71 let- 

 ters, giving 71 arrangements beginning with a ; the b may be fol- 

 lowed by each of the 71 letters, giving 71 arrangements beginning 

 with b, etc. So there are evidently 71 arrangements beginning 

 with each letter ; hence in all there are 7f arrangements of ti things 

 taken two at a time, allowing repetitions. 



Let us now find the number of arrangements, allowing repeti- 

 tions, o^ 71 things taken three at a time; and first to give definite- 

 ness to the ideas, consider the number of arrangements, allowing 

 repetitions, of four letters a, b, c, d taken three at a time. We 

 have written out the sixteen arrangements of four letters taken 

 two at a time, and now we may suppose each of these sixteen 

 arrangements to be preceded by the letter a, then each of these 

 sixteen arrangements to be preceded by b, etc. We then have 

 sixteen arrangements of three letters each, beginning with each 

 letter, and as there are four letters there are in all four times six- 

 teen, or sixty-four, arrangements of the letters a, b, c, d taken 

 three at a time, repetitions being allowed. 



Now, in the same way, if we have ;^ letters a, b, e, d, e,f, . . , 

 we may suppose each of the 7f arrangements two at a time to be 

 preceded by a, then each of these same ;^'' arrangements to be pre- 

 ceded by b, etc. 



Thus we get ;r arrangements beginning with a, 7f arrange- 

 ments beginning with b, 7f arrangements beginning with c, etc. 

 Hence in all we obtain 71 times 7^^ or ?z\ arrangements of 71 letters 

 taken three at a time, repetitions being allowed. 



hi geTieral, if we know the number of arrangements of 71 letters 

 taken 5 at a time, repetitions being allowed, we may find the num- 

 ber of arrangements of the n letters taken ^+1 at a time. 



Representing the number of arrangements, with repetitions, of 



